Quantum Delayed-Choice Experiment with a Beam Splitter in a Quantum Superposition

week ending PRL 115, 260403 (2015) PHYSICAL REVIEW LETTERS 31 DECEMBER 2015

Shi-Biao Zheng,1,* You-Peng Zhong,2 Kai Xu,2 Qi-Jue Wang,2 H. Wang,2 Li-Tuo Shen,1 Chui-Ping Yang,3 † ‡ John M. Martinis,4, A. N. Cleland,5, and Si-Yuan Han6,7 1Department of Physics, Fuzhou University, Fuzhou 350116, China 2Department of Physics, Zhejiang University, Hangzhou 310027, China 3Department of Physics, Hangzhou Normal University, Hangzhou 310036, China 4Department of Physics, University of California, Santa Barbara, California 93106, USA 5Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA 6Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA 7Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Received 11 August 2015; published 28 December 2015) A quantum system can behave as a wave or as a particle, depending on the experimental arrangement. When, for example, measuring a photon using a Mach-Zehnder interferometer, the photon acts as a wave if the second beam splitter is inserted, but as a particle if this beam splitter is omitted. The decision of whether or not to insert this beam splitter can be made after the photon has entered the interferometer, as in Wheeler’s famous delayed-choice thought experiment. In recent quantum versions of this experiment, this decision is controlled by a quantum ancilla, while the beam splitter is itself still a classical object. Here, we propose and realize a variant of the quantum delayed-choice experiment. We configure a superconducting quantum circuit as a Ramsey interferometer, where the element that acts as the first beam splitter can be put in a quantum superposition of its active and inactive states, as verified by the negative values of its Wigner function. We show that this enables the wave and particle aspects of the system to be observed with a single setup, without involving an ancilla that is not itself a part of the interferometer. We also study the transition of this quantum beam splitter from a quantum to a classical object due to decoherence, as observed by monitoring the interferometer output.

DOI: 10.1103/PhysRevLett.115.260403 PACS numbers: 03.65.Ta, 42.50.Dv, 42.50.St

The wave-particle duality is one of the fundamental the photon has passed through BS1. Therefore, the photon mysteries that lie at the heart of quantum mechanics. could not “know” in advance which behavior it should However, these two incompatible aspects cannot be observed exhibit. Wheeler’s delayed-choice experiment has been simultaneously, as captured by Bohr’s principle of comple- demonstrated previously [9–13], where the spacelike sep- mentarity [1–5]: Particlelike versus wavelike outcomes are aration between the setup selection and the photon entry selected by experimental arrangements that are mutually into the interferometer was achieved in Ref. [12]. Recently, exclusive. This is well illustrated by the Mach-Zehnder a quantum version of the delayed-choice experiment was interferometer, as shown in Fig. 1(a). Split by the first beam suggested [14] in which the action of BS2 is controlled by a splitter (BS1), a photon travels along two paths, 0 and 1. quantum ancilla. The scheme is illustrated in Fig. 1(b), The relative phase θ between the quantum states associated where each beam splitter is replaced by a Hadamard with these paths is tunable. In the presence of the second operation H, and the second Hadamard operation is condi- beam splitter (BS2), the two paths are recombined and the tionally applied following the phase shift θ. The quantum probability for detecting the photon in the detector D0 or D1 system exhibits wavelike behavior if the ancilla is in its j1i is a sinusoidal function of θ exhibiting wavelike interference state and the second Hadamard is applied; if the ancilla is fringes. On the other hand, in the absence of BS2,the instead in j0i, the second Hadamard is not performed, and experiment reveals which path the photon followed, and the system displays particlelike behavior. When the ancilla the photon is detected in one or the other detector with is prepared in a superposition of j0i and j1i, the result is a equal probability 1=2, thus behaving as a particle. superposition of wavelike and particlelike states, entangled One can argue that the behavior of the photon is with the ancilla [14]. This clearly precludes the system predetermined by the experimental arrangement, where knowing in advance which setup will be selected; the the presence or absence of the second beam splitter affects quantum superposition effectively replaces the need for the photon prior to its entering the interferometer. The Wheeler’s spacelike separation of the photon entry into the possibility of such a causal link is precluded in Wheeler’s interferometer and the measurement selection. Instead, the delayed-choice experiment [6–8], in which the observer wavelike and particlelike behaviors are postselected by randomly chooses whether to insert BS2, and thus whether measuring the ancilla in its ðj0i; j1iÞ basis. This allows the to perform an interference or a which-path experiment, after complementary wave and particle behaviors to emerge

(a) inequality; we note that these measurements are subject to the fair-sampling assumption: that is, the large number of coincident photon pairs that fail to be registered are assumed to obey the same statistics as those that are recorded. Here, we theoretically propose and experimentally (b) implement a variant of the quantum delayed-choice experiment, using a superconducting Ramsey interferometer and the process shown in Fig. 1(c); we note that the Ramsey and Mach-Zehnder interferometers are physically different, but they actually implement the same quantum circuit [20].In (c) our interferometer, a two-level quantum system (a superconducting qubit) with ground state jgi and excited state jei, which correspond to the paths 0 and 1 in the optical interferometer, is subjected to two Ramsey rotations, R1 and R2ðθÞ, which correspond to the two beam splitters ’ FIG. 1 (color online). (a) Schematic of Wheeler s delayed- of the Mach-Zehnder interferometer but are induced by choice thought experiment. A photon’s state is transformed by the microwave fields. The tunable phase shift θ between the first beam splitter, BS1, into a superposition of two paths, 0 and 1, followed by a tunable phase shift θ. While the photon is inside the two interfering paths is incorporated into the microwave ðθÞ interferometer, the observer decides whether or not to insert a field that produces R2 . R1 is generated by the microwave second beam splitter, BS2, thereby delaying the decision of field stored in a superconducting resonator. If the resonator whether to measure an interference pattern (wave aspect) or to is “filled” by exciting it into the appropriate microwave obtain which-path information (particle aspect), as revealed by the coherent state jαi, the qubit will undergo a Hadamard probability of detecting the photon at detector D0 and D1 as a transformation by interacting with this field. This rotates function of θ. (b) Schematic of the quantum delayed-choice the input state from jgi into a superposition of jgi and jei, experiment using a controlling quantum ancilla. The effect of producing two paths in the Hilbert space, as with the first each beam splitter is represented by a Hadamard operation, H, beam splitter in the Mach-Zehnder interferometer. These with the relative phase shift θ occurring in between these two two paths are then recombined by R2ðθÞ, and that recombi- operations. The second Hadamard is controlled by the ancilla, θ which can be prepared in a superposition state, allowing simulta- nation results in -dependent wavelike interference fringes neous preparation of the wave and particle aspects of the test in the probabilities of jgi and jei. If the resonator is instead system. (c) Schematic of the delayed-choice experiment, imple- “empty,” i.e., in its ground state j0i, no rotation R1 occurs, mented with a superconducting beam splitter that is put into a and the qubit remains in jgi. The second rotation R2ðθÞ then catlike superposition state. The test qubit, initially in the ground produces a superposition of jgi and jei, but the probabilities state jgi, is subjected to two operations, R1 and R2ðθÞ, equivalent of jgi and jei have no θ dependence, corresponding to in combination to the two Hadamard operations plus the phase particlelike behavior. We note that the required conditional θ shift . R1 is implemented (BS1 active) when the test qubit interacts dynamics cannot be realized when the order of the two with a superconducting resonator occupied by a classical coherent jαi ðθÞ rotations in Fig. 1(c) is inverted: To realize the resonator- field . R2 is implemented by a classical microwave pulse. j i The final probability of measuring the qubit in the ground state jgi state-dependent rotation, the qubit should be in the state g ’ or excited state jei, for active BS1, then depends on the relative before interaction with the resonator, so that the resonator s phase θ, thus exhibiting Ramsey interference. When ground state j0i cannot induce any rotation on the qubit; the superconducting resonator is instead in the ground state j0i, however, when the unconditional rotation R2ðθÞ is applied R1 is not implemented (BS1 inactive), and no θ-dependent Ramsey first, before the qubit-resonator interaction the qubit has a interference occurs. The wave and particle aspects of the qubit can probability of 1=2 to be excited by R2ðθÞ to the state jei,to be superposed by preparing the resonator in a quantum super- which the resonator’s ground state can produce a rotation jαi j0i position of filled and empty states, generating an output through the vacuum Rabi oscillation [21]. state that encodes both wavelike and particlelike behavior. The wave and particle behavior of the qubit can be investigated simultaneously by preparing the resonator in a from a single experimental setup, showing that the com- superposition of its filled jαi and empty j0i states, i.e., in a plementarity really resides in the experimental data, rather cat superposition state [22]. The nonclassical nature of this than resulting from the experiment’s physical arrangement. quantum beam splitter (QBS) can be revealed by measuring Quantum delayed-choice experiments have been per- the negative values of its Wigner function [23], which is the formed in both NMR [15,16] and optical [17–19] systems. resonator’s quasiprobability distribution in phase space. In the optical experiments reported in Refs. [18,19], the This enables one to verify the existence of a coherent quantum correlation between the test photon and the quantum superposition of the filled and empty, or active quantum ancilla was verified by the violation of a Bell and inactive, states of the QBS, without performing a Bell

260403-2 week ending PRL 115, 260403 (2015) PHYSICAL REVIEW LETTERS 31 DECEMBER 2015 measurement. In this case, after R2, the wave and particle Now we suppose that the resonator is instead initially states of the qubit are entangled with the state of the prepared in the cat superposition state resonator. This is in contrast to previous realizations of quantum delayed-choice experiments [15–19], where the jψ b;ii¼N ðcos φjαi − sin φj0iÞ; ð3Þ wave-particle response of the quantum system is deter- −jαj2 2 −1=2 mined by the state of a quantum ancilla, which is not itself a where N ¼½1 − e = sinð2φÞ . After the two rota- part of the interferometer but instead determines the action tions R1 and R2ðθÞ, the qubit-QBS system will then (or inaction) of one of the two beam splitters. In these approximately be in the entangled state experiments, the test system is entangled with the ancilla jψ i ≃ N ð φjψ ijαi − φjψ ij0iÞ ð Þ through the conditional action of the beam splitter, while qþb;f t cos w sin p ; 4 pffiffiffi the beam splitter remains a fully classical object. With our 2 N ¼½1 − −jαj =2 ð2φÞ 2−1=2 setup, we can also investigate the transition of the QBS to a where t e sin = . The probability j i classical beam splitter due to its intrinsic environmentally Pe for finding the qubit in the state e is then close to induced decoherence. As far as we know, this is the first 1 θ interference experiment in which one beam splitter in an ≃ N 2 2φ þ 2φ 2 ð Þ Pe t 2 sin cos cos 2 : 5 interferometer is in a coherent superposition of its active and inactive states. We note that our setup, using a temporally based Ramsey interferometer, does not permit The Ramsey interference pattern thus simultaneously ’ exhibits the wave and particle behaviors, through the the spacelike separation required for Wheeler s original θ gedanken experiment. presence and absence, respectively, of dependence in the two terms that make up Pe. The quantum coherence The rotation R1 of the qubit is produced by the micro- jαi j0i wave field stored in the resonator, which is resonantly between the two state components and precludes the j i↔j i possibility of the qubit knowing in advance what type of coupled to the qubit transition g e with coupling φ ¼ 0 strength Ω. If the resonator is in the state jαi, a coherent experiment will be performed. We note that for this microwave photon state with amplitude α and mean photon interferometer is equivalent to the cavity QED analogue 2 jei number hni¼jαj , the qubit exchanges energy with the described in Ref. [5] except there the qubit is initially in . The detailed implementation of this quantum delayed- resonator and Rabi oscillates between jgi and jei.For choice experiment involves a superconducting resonator simplicity, we assume α is real and positive. When α ≫ 1, and two tunable superconducting phase qubits, one qubit after an interaction time tα ¼ π=ð4pαΩffiffiffiÞ, the qubit state is serving as the test qubit and the second control qubit used approximately jψ αi¼ðjgi − ijeiÞ= 2, with the resonator to program the state of the superconducting resonator, as field left close to its original state jαi. Numerical simulation well as to read out the resonator at the end of each shows that the approximation is good even for moderate experiment. The two qubits are coupled to the resonator values of α. This result has a simple qualitative explanation: with on-resonance coupling strengths Ω and Ω0, respec- When the resonator’s coherent field is not too weak, tively, and the effective coupling of each qubit to the its Poissonian photon-number distribution—and hence its resonator can be switched on or off by tuning the qubit on state—is insensitive to a one-photon change. If the reso- or off resonance with the resonator. The experimental nator is instead in its ground state j0i, the rotation R1 does apparatus is identical to that described in Ref. [24]. not occur and the qubit remains in jgi. The second pulse, Using this apparatus, we can arrive at the state jψ b;ii of R2ðθÞ, generated by a classical microwave pulse, performs pffiffiffi Eq. (3) with α ¼ 2, and arbitrary values of φ (Ref. [21]; see j i → ðj i − −iθj iÞ 2 j i → the transformationspffiffiffi g g ie e = and e the Supplemental Material [25]). For φ ¼ π=4, the fidelity ðj i − iθj iÞ 2 jψ i e ie g = . When acting on the state α , this F ¼hψ b;ijρb;ijψ b;ii¼0.726 0.028 (ρb;i is the density results in the state operator corresponding to the output state). It would be preferable to obtain states with larger amplitude α,butwe jψ i¼− ½ ðθ 2Þ iθ=2j iþ ðθ 2Þ −iθ=2j i ð Þ w i sin = e g cos = e e : 1 are limited by the small nonlinearity of our phase qubits to α ≲ 2 The probability of measuring the qubit in jei is then . Fortunately, numerical simulations show that the 2 overlap between the final qubit-QBS state and jψ þ i of cos ðθ=2Þ, showing the θ-dependent interference associ- q b;f Eq. (4) is higher than 0.98 even for this value of α (ignoring ated with wavelike behavior. If R1 did not occur, due to the experimental imperfections). After generating jψ i,we microwave resonator being in its ground state j0i, the final b;i resonantly couple the test qubit to the resonator for a time state after the second rotation, R2,is pffiffiffi tα, and then apply a carefully tuned, on-resonance micro- −iθ wave π=2 pulse to the qubit, thus implementing jψ pi¼ðjgi − ie jeiÞ= 2: ð2Þ the rotation R2ðθÞ. Figure 2(a) displays the measured j i j i 1 2 The probability to be in g or e is = , representing probability Pe as a function of θ and φ, which clearly particlelike behavior without the θ-dependent interference demonstrates the morphing between particle and wave effects. behavior. To more precisely understand this behavior, we

260403-3 week ending PRL 115, 260403 (2015) PHYSICAL REVIEW LETTERS 31 DECEMBER 2015 need to examine the state of the QBS. This is achieved after the interaction, owing to the photon-number depend- using the control qubit, initially in the state jg0i, and tuning ence of the Rabi frequency. These two reasons account for 0 it into resonance with the resonator for a time π=ð2αΩ Þ.If the θ-dependent interference effect in Pe;g0 , with a fringe the resonator contains a coherent field jαi, the control qubit contrast measured to be 0.19. It is noted that taking into will absorb a photon and undergo the transition jg0i → je0i, account the imperfect initial state as we prepared exper- while if the resonator is in j0i, the qubit remains in jg0i. imentally, we could verify that the measured fringe data The test qubit behavior is postselected by correlating its (dots with error bars) are in good agreement with the measurements with the outcomes of the control qubit numerical simulation (lines) in Fig. 2(b). With an increase measurements. We note that jαi is not strictly orthogonal in the coherent state amplitude α, the state jαi is more clearly to j0i, with the overlap jhαj0ij2 ≈ 10−2 for α ¼ 2, so that distinguished from j0i, and as a result the unwanted these two components cannot be unambiguously discrimi- oscillations decay dramatically: For α ¼ 3, the calculated nated; there is a small probability that the detection of j0i fringe contrast is reduced to 0.048. actually comes from jαi. With α somewhat larger, this It should be noted that the same statistical data can be overlap will become negligible: For α ¼ 3, the overlap is produced if the resonator field is in the classical mixture 2 2 only about 10−4. cos φjαihαjþsin φj0ih0j. To exclude the classical inter- jαi Figure 2(b) shows the measured probabilities Pe;e0 and pretation, the existence of quantum coherence between 0 and j0i should be verified. The quantum state of the Pe;g for detecting the test qubit in the state jei conditioned upon the detection of the control qubit in je0i and jg0i, resonator field can be characterized by measuring its ðχÞ respectively; these are measured as a function of θ. Here the Wigner function (WF) W , which describes the quasi- parameter φ is π=4, corresponding to the QBS being in an probability distribution of the microwave field in resonator equal superposition of its active and inactive states. As phase space [23]. The WF associated with the density ρ 0 operator b is defined as expected, Pe;e exhibits Ramsey interference fringes, with the contrast reaching 0.839, while Pe;g0 remains almost 2 iπa†a 0 WðχÞ¼ Tr½e Dð−χÞρ DðχÞ; ð6Þ constant. The slight oscillations in Pe;g are mainly due to the π b fact that the amplitude of the coherent state component jαi is somewhat small, so that the control qubit has a small where χ is the (complex) coordinate in phase space and D is probability of remaining in the ground state after the the displacement operator. This quantity is always non- interaction even when the resonator is in jαi. On one hand, negative for the QBS in a classical mixture; the observation the coherent state contains a vacuum state component, which of negative values in regions of phase space is a signature of is decoupled from the qubit state jg0i. On the other hand, not quantum interference. The Wigner function of the QBS was all the other superposed Fock states can make the control measured using the control qubit, following the procedure qubit flip to the excited state je0i with a unity probability developed in Ref. [21]. In Fig. 3(a), we display the WF after performing the second rotation R2ðθÞ, but without reading out the state of the test qubit, while in Figs. 3(b) and 3(c) we (a) (b) show the WFs measured with the test qubit having been measured in jgi and jei, respectively, all for θ ¼ π=2 and φ ¼ π=4. Simulated and measured WFs are shown in the upper and lower panels, respectively. Experimental imperfections are not included in the numerical simulations. Since the qubit wave state jψ wi is not orthogonal to the particle state jψ pi, there is some quantum coherence between jαi and j0i, even when the test qubit is traced out. As a result, the WF exhibits a strongly nonclassical feature around χ ¼ 1, as shown in Fig. 3(a). The measured WF has a minimum value of −0.258 0.030 at χ ¼ 0.84–0.03i. In Fig. 3(a), the shapes of the calculated FIG. 2 (color online). Measured Ramsey interference signal. and measured WFs agree well, demonstrating that the (a) Probability Pe, defined in Eq. (5),vsθ and φ. The amplitude jψ i α ¼ 2 measured negative quasiprobabilities are due to quantum of the coherent state component of the cat state b;i is . interference between jαi and j0i. The existence of quantum The transition between the wave and particle behaviors is clearly coherence between these two state components implies that apparent. (b) Probabilities Pe;e0 and Pe;g0 ,vsθ, for detecting the test qubit in jei given that the control qubit is detected in je0i and their correlation with the behavior of the test qubit shown jg0i, respectively. The control qubit’s state reflects the QBS state, in Fig. 2(b) is nonclassical. Without reading out the test ’ θ as this qubit is measured after it has interacted with the post-R2 qubit s state, the WF is independent of the value of since resonator state for a time π=ð2αΩ0Þ. Data are for φ ¼ π=4. Error any local unitary operation on the test qubit does not affect bars indicate the statistical variance. Lines are simulations taking the QBS state after their interaction. When the test qubit’s into account imperfections in preparing jψ b;ii. state is measured, the minimum value of the WF becomes

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(a) (b) (c) the size and fidelity of the cat state and to explore the gradual transition from a quantum to a classical measuring device. This experiment was realized using a circuit QED system; however, similar experiments could be performed using microwave cavity QED [31,32] or ion-trap setups [33,34]. We further note that the idea of producing a conditional rotation on a qubit with an oscillator in a superposition state may be useful for the generation of important entangled states. These conditional dynamics, together with the qubit- oscillator quantum state transfer [35], could be used to produce entanglement between two oscillators. Another example is the generation of entanglement between multiple qubits and an oscillator by performing rotations on these ’ FIG. 3 (color online). Wigner tomography of the quantum beam qubits conditional on the oscillator sstate. α ¼ 2 θ ¼ π 2 φ ¼ π 4 splitter. The parameters are , = , and = . The We thank D. J. Wineland for the fruitful discussions. quantum beam splitter Wigner functions (WFs) are displayed for three cases: (a) the test qubit state is not read out; (b) the This work was supported by the Major State Basic test qubit is measured in jgi; (c) the test qubit is measured in jei. Research Development Program of China under Grants The simulated and measured WFs are shown in the upper and No. 2012CB921601, No. 2014CB921200, and No. lower rows, respectively, as a function of the coordinate χ in 2012CB927404, and the National Natural Science resonator phase space. Experimental imperfections are not Foundations of China under Grants No. 11374054, included in the numerical simulations. The minimum values of No. 11222437, and No. 11374083. 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